Sure, let's go through the process of subtracting these two fractions step by step.
The original problem is:
[tex]\[ \frac{5}{x^2 - 6x + 8} - \frac{2}{x^2 - 4x + 4} \][/tex]
1. Factor the denominators, if possible:
- The first denominator [tex]\(x^2 - 6x + 8\)[/tex] factors to:
[tex]\[ x^2 - 6x + 8 = (x - 2)(x - 4) \][/tex]
- The second denominator [tex]\(x^2 - 4x + 4\)[/tex] factors to:
[tex]\[ x^2 - 4x + 4 = (x - 2)^2 \][/tex]
2. Express both fractions with their factored denominators:
[tex]\[ \frac{5}{(x - 2)(x - 4)} - \frac{2}{(x - 2)^2} \][/tex]
3. Find a common denominator for the two fractions:
- The common denominator is obtained by taking the least common multiple (LCM) of the two denominators [tex]\((x - 2)(x - 4)\)[/tex] and [tex]\((x - 2)^2\)[/tex]. The LCM in this case is [tex]\((x - 2)^2(x - 4)\)[/tex].
4. Rewrite each fraction with the common denominator:
- For the first fraction:
[tex]\[ \frac{5}{(x - 2)(x - 4)} = \frac{5(x - 2)}{(x - 2)^2(x - 4)} \][/tex]
- For the second fraction:
[tex]\[ \frac{2}{(x - 2)^2} = \frac{2(x - 4)}{(x - 2)^2(x - 4)} \][/tex]
5. Subtract the fractions:
Now we can subtract the two fractions because they have a common denominator:
[tex]\[ \frac{5(x - 2)}{(x - 2)^2(x - 4)} - \frac{2(x - 4)}{(x - 2)^2(x - 4)} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ \frac{5(x - 2) - 2(x - 4)}{(x - 2)^2(x - 4)} \][/tex]
6. Simplify the numerator:
- Distribute the constants in the numerator:
[tex]\[ 5(x - 2) - 2(x - 4) = 5x - 10 - 2x + 8 \][/tex]
- Combine like terms:
[tex]\[ 5x - 2x - 10 + 8 = 3x - 2 \][/tex]
7. Write the simplified result:
[tex]\[ \frac{3x - 2}{(x - 2)^2(x - 4)} \][/tex]
The subtraction of the two original fractions gives us:
[tex]\[ \frac{3x - 2}{(x - 2)^2 (x - 4)} \][/tex]
And we can express this as:
[tex]\[ \frac{3x - 2}{x^3 - 8x^2 + 20x - 16} \][/tex]
This is the final answer for the problem:
[tex]\[ \frac{3x - 2}{x^3 - 8x^2 + 20x - 16} \][/tex]
